Initialized fractional calculus

The composition law of the differintegral operator states that although:

$\backslash mathbb\{D\}^q\backslash mathbb\{D\}^\{-q\}\; =\; \backslash mathbb\{I\}$

wherein D^−q is the left inverse of D^q, the converse is not necessarily true:

:$\backslash mathbb\{D\}^\{-q\}\backslash mathbb\{D\}^q\; \backslash neq\; \backslash mathbb\{I\}$

Consider elementary integer-order calculus. Below is an integration and differentiation using the example function $3x^2+1$:

:$\backslash frac\{d\}\{dx\}\backslash left[\backslash int\; (3x^2+1)dx\backslash right]\; =\; \backslash frac\{d\}\{dx\}[x^3+x+C]\; =\; 3x^2+1\backslash ,,$

Now, on exchanging the order of composition:

:$\backslash int\; \backslash left[\backslash frac\{d\}\{dx\}(3x^2+1)\backslash right]\; =\; \backslash int\; 6x\; \backslash ,dx\; =\; 3x^2+C\backslash ,,$

Where C is the constant of integration. Even if it was not obvious, the initialized condition ƒ'(0) = C, ƒ(0) = D'', etc. could be used. If we neglected those initialization terms, the last equation would show the composition of integration, and differentiation (and vice versa) would not hold.

Working with a properly initialized differ integral is the subject of initialized fractional calculus. If the differ integral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, information is lost, as with C in the first equation.

However, in fractional calculus, given that the operator has been fractionalized and is thus continuous, an entire complementary function is needed. This is called complementary function $\backslash Psi$.

:$\backslash mathbb\{D\}^q\_t\; f(t)\; =\; \backslash frac\{1\}\{\backslash Gamma(n-q)\}\backslash frac\{d^n\}\{dt^n\}\backslash int\_0^t\; (t-\backslash tau)^\{n-q-1\}f(\backslash tau)\backslash ,d\backslash tau\; +\; \backslash Psi(x)$

• Initial conditions
• Dynamical systems

• {{citation|title=Initialized Fractional Calculus|last1=Lorenzo|first1=Carl F.|last2=Hartley|first2=Tom T. |year=2000 |publisher=NASA|url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20000031631.pdf|ref=none}} (technical report).

Category:Fractional calculus